How are emergent constraints quantifying uncertainty and what do they leave behind?

The use of emergent constraints to quantify uncertainty for key policy relevant quantities such as Equilibrium Climate Sensitivity (ECS) has become increasingly widespread in recent years. Many researchers, however, claim that emergent constraints are inappropriate or even under-report uncertainty. In this paper we contribute to this discussion by examining the emergent constraints methodology in terms of its underpinning statistical assumptions. We argue that the existing frameworks are based on indefensible assumptions, then show how weakening them leads to a more transparent Bayesian framework wherein hitherto ignored sources of uncertainty, such as how reality might differ from models, can be quantified. We present a guided framework for the quantification of additional uncertainties that is linked to the confidence we can have in the underpinning physical arguments for using linear constraints. We provide a software tool for implementing our general framework for emergent constraints and use it to illustrate the framework on a number of recent emergent constraints for ECS. We find that the robustness of any constraint to additional uncertainties depends strongly on the confidence we can have in the underpinning physics, allowing a future framing of the debate over the validity of a particular constraint around the underlying physical arguments, rather than statistical assumptions.